So im trying to learn some basic logic and I got stuck on how we use implications with the universal quantifier.
My problem is the following:
$\forall x {\in}\varnothing\; P(x)$ is the same as saying
$\forall x\; \big(x\in\varnothing \to P(x)\big)$
However if I've been given a statement of the form $\forall x\in A \; \big(x\in\varnothing \to P(x)\big)$ should I read this as $\forall x\; \big(x\in A \to x \in \varnothing \to P(x)\big)$
or instead $\forall x\; \big(x\in A \wedge x \in \varnothing \to P(x)\big)$.
Thanks in advance .
A statement of the form $$\forall x\in A[Q(x)]$$ always means $$\forall x[x\in A\rightarrow Q(x)],$$ by definition (the first statement is just an abbreviation for the second). So, $$\forall x\in A[x\in\emptyset \rightarrow P(x)]$$ means $$\forall x[x\in A\rightarrow (x\in\emptyset\rightarrow P(x))].$$ This is equivalent to $$\forall x[(x\in A\wedge x\in\emptyset)\rightarrow P(x)]$$ though since $x\in A\rightarrow (x\in\emptyset\rightarrow P(x))$ and $(x\in A\wedge x\in\emptyset)\rightarrow P(x)$ are equivalent. I'm not sure if this is what you meant with your second statement since you left out the parentheses.